Method and apparatus for analyzing shape and bending stresses in structural elements



- o. G. BOWEN ,7 METHOD AND APPARATUS FOR ANALYZING SHAPE AND BENDING STRESSES IN STRUCTURAL ELEMENTS Filed Feb. 15, 1950 July 28, 1953 [N VENTOR. Our/1? GERALD BOWN BY H/s HTTO RNE. v5. HARRIS, AlL-icH, F05 TE)? 5: HARRIS Patented July 28, 1953 METHOD AND APPARATUS FOR ANALYZING SHAPE AND BENDING STRES SES IN STRUCTURAL ELEMENTS Oliver Gerald Bowen, Los Angeles, Calif., assignor to Presan Corporation, Los Angeles, Calif., a corporation of California Application February 13, 1950, Serial No. 144,014

v 11 Claims. 1

My invention relates to a method and apparatu for analyzing or determining the shape of a loaded element and more particularly for analyzing the bending stresses in an element whenloaded.

It is an important object of the present in. vention to provide a novel method and apparatus employing optical principles and which can be used in the determination or analysis of shape or bending stresses in an element of a structure. typically'a flat element such as a floor slab sub ject to bending due to the application of transverse loading.

The invention will be herein described with reference to the solution of a relatively simple problem, namely, the analysis ofbending stresses in a reinforced-concrete'floor slab of a rectangular building, the slab being supported by the four side walls and by a central column. This example has been chosenmerelybecause of simplicity in explaining the principles of the invention. While the design of such a floor slab is relatively simple when employing known engineering formulas, the determination of the actual bending moments in a more complex building structure becomes extremely tedious and difficult. In certain irregularly-shaped or irregularly-supported floor slabs, use of conventional formulas and methods becomes practically impossible except to give rough approximations as concerns the placement and size of reinforcing rods.

For example, the determination of the actual bending moments in a monolithic floor system which spans in more than one direction, represents an extremely diflicultproblem, particularly if the spacing of the supports or column is irregular or non-uniform.

It is an object of the invention to provide a novel method and apparatus well suited to the solution of such a problem, as well as to the solution of problems involving stresses in other structural elements having flat surfaces and which may be irregularly supported to establish irregular stress patterns upon the application of transverse or other loads whether of a uniform or concentrated character.

It is another object of the present invention to provide a simplified method and apparatus for analyzing stresses in an irregularly supported element of a structure, involving the use of a .model of such structure. of the same relative stiifness as the structure and should provide an element corresponding to the element of the structure to beanalyze d or designed. The stress analysis can be made on the basis of the flexure-o f such a glslllgnfiof the The model should be 2" I model or on the basis of flexure of the corresponding element of a completed or partially completed structure. Hereinafter, when speaking of such an element, it should be understood that the element may be a part of such a model or a part of such a structure. T

It is an object ofthe present invention to analyze the shape or bending stresses in an element by making reflective or by mirrorizing (the latter term being used in a broad sense) the entire area of the element or some relatively large portion of the entire area which is to be critically examined. Onto this reflective surface is transmitted an optical pattern of known design to obtain a reflected optical pattern. When the element is loaded, the reflected optical pattern will differ from the transmitted optical pattern as a function of the bending of the element under load. If the element has a plane reflective surface, comparison of the transmitted and reflected optical patterns in corresponding zones will give a measure of the deformation and the resulting bending stresses in the element in a corresponding zone. The invention has among its objects the application of such principles for the purposes heretofore set forth.

If the reflective surface is not exactly plane, either because of surface irregularities or an overall curvature under no-load condition, the reflected no-load optical pattern will deviate somewhat from the transmitted optical pattern. However, a further change in the reflected optical pattern will result upon loading of the element and it is within the scope of the present invention to compare the no-load reflected optical pattern with the optical pattern obtained under loaded condiions to analyze the shape or bending moments in the element under consideration.

A general object of the invention is to provide amethod and apparatus for the relatively rapid and exact analysis of stresses in complex structures by photographic or optical means.

Further objects and advantages of the invention will be evident to those skilled in the art from the following description, taken from the drawing, which illustrates and describes the use of the invention in solving a relatively'simple problem, yet which will. exemplify and guide those skilled in the art to the use of the invention in solving the more complex problems for which it is particularly adapted.

Referring to the drawing:

Fig. 1 is an iosmetric view of a simple arrangement of equipment contemplated by the invention;

Fig. 2 is a side view of the equipment of Fig. 1, with the patterned screen shown in section;

Fig. 3 is a simplified view of one type of patterned screen or. optical pattern that 'may be employed{ and Fig. 4 is a diagrammatic view of the type of distorted optical image produced under load conditions with the equipment shown in Fig. l,

' Referring particularly toIFig. -,'iu1e e1 mem to be analyzed is indicated by the numeral [9,, shown as a part of a model [I of a building structure, although it may be an element 6f an actual building structure. By "way o'f example',

the element will be considered as an element of the model ll, built to the same. relative. stiffness as a building structure in which the corresponding element is a monolithic floor -slab.

As suggested in Fig. 1, the element It! is a fiat el'en'ient having a tram reflective 'or mirro'rized surface I 3and'a back' surface M. The element (0 is supported at it's opposite' sides by sidewalls l and' l"6 and at its ends'by end" walls l1- and [8, corresponding respectively to side walls and end walls of a rectangular building. structure. The walls l5 i8 are respectively secured to and supported by a thicker foundatio'nwall 2i), corre-;

sponding to' the fdundationof-the. building" structure. "Fbrpurpse of i-llustration, a single col umri 23' extends between and is secured to the surface ld-"bfthe element: W and thefoundation Wall i,

" The model II is constructed of structural elements or the same relative stiffness i the building structure "It is also 'a model; er the buildin structure in the sense of being-proportional in shape and placementofsupports'an'd loads' In an actual situation, the supportsmay include a largenumb'erof irregiilarly' positioned columns, baams, etcl'fpla'cedf ih amodl' 'or building or otherstructure of regular 'or' irreg ular shape.

The "ad er {I may bemadaof any suitable material but the element {0 shbuld'be of amaterial which-can"bemade'reflectiveso as to present a smooth reflective surface hereinbefo're designated as the" reflective siurface' I31" While metals can beusedforthe eiemem [0, they are notthe best materials, particularly if they present some what porous mates? Glass ean'te usedbut the: most convenient and satisfactory materials are the so-called plastic'mat" ia'lsj "The-material known as Lucite',"f'or example, is; excellently suited, being of low porosityand being capable of being highly polished to rec ive s vapordeposited coating tofcriri the reflective surface [3. The elementlil' of tlimodel'FF sl-iouldpref erably be of a material having the" following characteristics: (1) It should have a relatively high ultimate tensile strengttiass'compared; with its Youngs. modulus. (2.) It. should be, capable f bein equately'p l-ished to. be made. eifee tive. (3) It should be, capable, of being joined into. a monolithic structure. (,4) ltsstress strain curve should be. close, to a straight line Within the range of. workin stresses. (5). It should have a reasonably 'fiatoptical surface in the as-purchased condition or befcapable of optical finishing. (6) It should be reasonably stable ascorl Gems. shrinkage,v plastic flow and temperature effects.

The walls l5- l8, and 20, may, be formedof the same material asthe element I l], and, the same, may be true as ta thelcolumn it illustrated. In practice, the complete model l l may be mad? from sheets or rods of transparent plastic material adhered together by a suitable solvent or adhesive to provide a hollow model having an inner space 24 bounded and closed by the surface 14 of the element [0 and by the inner surfaces-of' the walls [5-18 and 29.

If a uniform transverse load is to be applied to the element [0, it is one of the desirable features of the invention that the model H or the inner space 24- should be made air tight so that a p'ressurediiference can be established inside and outside the model. This is a simple and extremely accurate way of applying a uniform load, although it should be clear that the invention isnot limited thereto and that uniform loads can be applied to the element ii) by other means, suchasby partially filling the inner space 24 with shot or liquid and permitting the weight thereof to deform the element 10 between its supports; also, air-tight bags may be inserted in such a manner. as to give a uniform load over. only'part of the element It.

- As shown in Fig. l, a difference in pressure can be established inside and outside the model by. forcing air into or evacuating air'from the inner space 24' through a' pipe 25; the pressuredifierence'between the interior'and exterior of'the model being shown by a manometer 21-. In practice, an internal pressure of a few inches of water can be maintained inside the model to cause the element I0 to bow outwardly between its supports. The resulting distorted reflected optical pattern can be used to determine the curvature of or stress in the element Hi just as effectively as if the inner space 2 1 is at. subatrnospheric pressure as the degree and location of the flexure of theelement 10 will be the same, except reversed, in the two instances In the simplestpractice of the invention, the model H is placed in front of a source of an optical pattern shown as a patterned screen having 'an opening 3! therein. The screen 30 is shown as having a front surface 32- facing the model Ii and carrying a pattern 33 of known design. In the preferred practice of the invention; the' pattern is a regular arrangement of parallel lines, usually with'crossing lines to form a grid, as suggested in Fig. 3. Use of the invention is made simpler if the longitudinal and transverse lines are equally spaced. However, any optical pattern of known design, preferably a two-dimensional pattern, can be transmitted to the reflective surface 13' without departing from the spirit of the invention. This pattern may even be irregular or it'may consist merely of a large number of dots or iines arranged in a known pattern. The use of 'a pattern involving straight lines is preferred and the use of the invention is facilitated if" the direction ofthe lines corresponds to the plane or planes of the model H or the building structure in which the stresses are to be analyzed.

While a back-lighted translucent screen 30 can be used, I prefer to use an opaque screen which is front-lighted by a plurality of' reflactors 35 carrying light sources and designed equally to illuminate the front surface 32 of the screen 30. In any. event, an optical pattern of known design is transmittedto the reflective surface [3 and reflected therefrom as a reflected optical pattern which can be viewed through the opening 3| of the screen. In the preferred practice of the invention, the reflected optical pattern is, recorded photographically and for this purpose, I prefer to dispose a' camera 40 to "the rear of the screen 30 and having a lens 4| receiving the reflected optical pattern to focus same on a photographic film or plate 42 at the rear of the camera. In other instances, a lens 4| may be mounted in the opening 3|, the screen 30 forming one wall of a dark room and focusing the image on a photo-sensitive or other surface of the dark room on the opposite side from the screen 30. Other systems, sometimes employing auxiliary mirrors, can be used if it is not desired that the opening 3| be employed in the screen to receive the reflected image.

If the reflective surface I3 is plane and if the element [0 is not loaded, it will beapparent that an optical pattern of known design will be transmitted to the reflective surface 13 and will be reflected therefromas an optical pattern of the same design, this pattern being accurately recorded by the camera 48. If the reflective surface I3 has surface irregularities, the optical pattern reflected from the unloaded element Ill will deviate from the transmitted pattern in minute degree, depending upon the surface irregularities. Such a deviating pattern will be recorded by the camera 40 and the resulting photographic image or an enlargement thereof can be used for purpose of comparison with the load-distorted pattern later recorded.

If the element H is loaded, as by applying a superatmospheric pressure inside the model, the resulting curvature of the reflective surface [3 will distort the reflected optical pattern, the distorted pattern being recorded by the camera All. By comparing the distorted photographic image, further enlarged if desired, with the transmitted pattern or the reflected no-load pattern, the degree of flexure of the element [0 in any particular zone or area can be observed and the stresses or bending moments in such zone determined.

By way of example, if the transmitted optical pattern represents a grid of two series of equallyspaced lines as suggested in Fig. 3 (the actual number of lines being much greater than that exemplified) the reflected distorted optical pattern will show curved or differently spaced lines in the stressed zones of the element II] (the word stressed is usually herein used in a broad sense as including stress or strain forces). In any particular zone, measurement of the difference in spacing as between the lines of the transmitted and reflected patterns will give a basis for calculation of the actual bending moment in this zone. By determining the bending moments throughout the element In or along selected lines therein, the stress pattern of any portion or all of the element can be plotted. From this, one can determine, in the case of reinforced concrete, for example, the reinforcing required to resist the bending stresses and can design the number, placement, length and shape of the reinforcing rods or members required for adequate support of the transverse loading.

For example, the flexure of the element I0 shown in Fig. l and when an internal superatmospheric pressure is applied to the model H, will produce a distorted optical pattern of the type suggested very diagramatically in Fig. 4. In the zones of highest stress, the parallel lines of the transmitted optical pattern will show the greatest deviation in spacing, and measurement of this deviation along any selected line A--A, as compared with the lines of the transmitted image, will permit calculation of the actual bending moments along this line The bending moment at'any position or zone can be determined from the following formulae:

direction and py in the 'y direction; C is a constant depending on the geometry of the equipment; JX is the distance in the x direction between grid lines on the photographic image in the zone in question when the element is not,

stressed; Kx is the distance in the .r direction between such lines when the element is stressed? My is the bending moment in the y direction; ,u is :Poissons ratio; E is the Youngs modulus; and I is the moment of inertia. The ratio is thus a measure of the extent and direction" When this ratio isof the bending moment. greater than one a negativemoment is present and when less than one a positive moment is present. Stated in other words and with an internally-pressured model, as the lines become farther apart in the strained condition a nega-' If an extremely accurate grid of lines is pres:

ent on the screen 30 and the unstressed reflective surface I3 is plane, the photographic image of the distorted pattern can be correspondingly enlarged and the line spacings measured along a line of the model or structure where the mo ment measurement is to be taken. If the grid of lines is less accurate or if there are small irregularities in the unstressed reflective surface l3, a photographic image of the pressurally-undistorted or no-load reflected image can be made, enlarged and compared, along any line where moments are to be determined, with the similarly enlarged photographic image showing the lines as spaced when the element I 0 is under load. This can be done by means of a comparison microscope.

In either event, the calculated bending moments along any such line can be plotted, with positive moments above the X-axis and negative moments below such axis, to produce an ini-.' tial bending moment diagram which may be sufficiently accurate for some design purposes; If desired, a similar bending moment diagram can be plotted from the unstrained image and employed to correct the initial bending moment diagram. v

In determining the amount and placement of steel reinforcing needed in a reinforced concrete slab, represented by the element IU of the model, such moment curves are of great value. The steel in the top and bottom of the slab, required to resist negative and positive bending moments, respectively, will be shown by the portions of the curve below and above the X-axis thereof. Alternatively, a contour-like plot of bending moments can be made and used'for design of the amount, placement and length of the reinfcrcing'members.

While the invention has been described with reference to slab design, it is also applicable to the stress analysis in roofs, flat or pitched, or other flat elements and can sometimes be applied to other elements of a non-flat nature. Likewise, while a particular means has been described for creating and photographing the optical pattern, it. should be clear that the optical pattern of known design can-be transmitted to or received by the reflective surface from varioussources and in various ways and that the reflected pattern or its pattern can be viewed, photographed or compared with the pressurallyundistorted pattern in various ways without departing from the spirit of the invention.

I claim as my invention:

1. A method of analyzing the bending stresses in a substantially flat element of a structure, said substantially flat element being supported at predetermined pointsand being subject to bending due to the application of transverse loading, which method includes the steps of: building a model of said structure, said model having a correspondinglyeshaped, correspondingly supported substantially flat element having an exposed substantially flat surface; making said exposed substantially flat surface reflective; transmitting to said reflective surface to be reflected therefrom an optical pattern of predetermined design, the reflected optical pattern deviating from the transmitted optical pattern upon deformation of said reflective surface; applying a transverse load to said substantially fiat element: of said model perpendicular to and to deform said' reflective surface in degree and location determined by its supports and by such transverse load; and measuring the deviations of the resulting. reflected optical patterns as a measure of the deformation of and the resulting bending stresses in said substantially flat element of said structurewhen subjected to a corresponding transverse load.

2. A method as defined in claim 1, in which said transverse load applied to said substantially flat element of said model is produced by developin a pressure differential on opposite sides of saidsubstantially flat. element of said model.

3.,A method as defined in claim 1, in which said model is hollow and includes an inner space closed; by one, surface of said substantially flat element. of said model, the other surface of such substantially flat element being said reflective surface, and in which said transverse load is applied to. said substantially flat element of said model by developing apressure difierential between said surfaces of said substantially flat element. of said model.

4. A method as defined in claim 3, in which said inner space is substantially air-tight and said reflective surface is exposed to atmospheric pressure, and in which said pressure differential developed by changing the pressure in said inner space relative to atmospheric pressure.

A. method of determining the distribution and: magnitude of bending stresses in a structure, ccmprising the steps of providing a test member with a specular reflective; surface of substantial area, sup-porting spaced portions of said substantial area ofsaid test member against movement in adirectionperpendicular to said surface, loading said test member. by applying pressure thereto a direction. perpendicular to said surface and between said supported. portions to distortsaid 8. reflective surface, reflecting a single unitary image of a flxed pattern of geometrically related spaced indicia from said substantial area of said distortedsurface, and measuring the relative displacement of a plurality of said indicia in said reflected imageas an indication of the distribution and magnitude of the-stresses produced in corresponding portions of said substantial area by said loading.

6. A method as defined in claim 5 including the step of reflecting a first image of said pattern from said surface before applying pressure thereto to establish a subjective datum image, and wherein'said measuring step comprises measuring the differences in spacing between corresponding indicia on said datum image and the image reflected from said distorted surface as a measure of the distortions in corresponding portions of said surface caused by said loading.

7.. A method as defined in claim 5 including the step of photographically recording said reflected. image and wherein said measuring step comprises determining the difference in spacing between corresponding indicia on said recorded image and said flxed pattern.

8. A method. of determining the distribution and magnitude of bending stresses in a structure, comprising the steps of: providing a test member with a specular reflective surface of substantial area, supporting spaced portions of said substantial area of said test member against movement in a direction perpendicular to said surface, reflecting a single unitary image of a fixed pattern of geometrically related spaced indicia from said surface, making a. first permanent datum record of said reflected image, loading said test member by applying pressure thereto in a direction perpendicular to said surface and between said supported portions to distort said reflective surface, reflecting a second unitary image of said pattern from said distorted surface, making a second permanent record of said seccnd image, and determining the difference in spacing between a plurality of corresponding indicia on said first and second records as an of the distribution and magnitude of stresses in that area of said test member from. which said indicia were reflected.

9. In a system for determining the distribution and magnitude of bending stresses in a structure, the arrangement comprising: a member having a specular reflective surface of substantial area, spaced supports engaging said member and holding spaced portions of said area against move ment in a direction perpendicular to said reflective surface, means defining a fixed pattern of geometrically related spaced indicia, an image receiving means fixed-1y supported in position to receive an image reflected from said surface, means holding said member and supports in position to reflect a unitary image cf said pattern from said reflective surface to said image receiving means, and loading means arranged to apply a loading force to said member in a direction perpendicular to said reflective surface between said supports to distort said surface and thereby cause reflection of a distorted image of said pattern to said image receiving means.

10. Apparatus as defined in claim 9, wherein said means defining said patcrn comprises a patterned screen substantially parallel to said reflective surface, said screen having an opening therethrough, said image receiving means being positioned to receivev a reflected image of said pattern through said opening.

11. Apparatus as defined in claim 9, wherein Number Name Date said member comprises a wall of a hollow body 1,590,532 Lenouvel June 29, 1926 and wherein said loading means comprises means 2,014,688 Maddoux Sept. 17, 1935 for producing different fluid pressures within 2,380,501 Christian et a1 July 31, 1945 said hollow body. 5 2,459,418 Ellis Jan. 18, 1949 OLIVER GERALD BOWEN. FOREIGN PATENTS Number Country Date References Cited in the file of this patent 573,656 Germany {A m 5, 1933 UNITED STATES PATENTS 10 816,087 France p 1937 Number Name Date OTHER REFERENCES;

1,457,209 Chanier May 29, 1923 Scientific American, October 1946, page 178. 1,552,450 Roach i- Sept. 8, 1925 (Copy in U. S. Patent Oifice Library.) 

